Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Document related concepts

Euclidean geometry wikipedia , lookup

Line (geometry) wikipedia , lookup

Technical drawing wikipedia , lookup

Four-dimensional space wikipedia , lookup

Space wikipedia , lookup

History of geometry wikipedia , lookup

Geometrization conjecture wikipedia , lookup

Transcript
```Statistics
of Anatomic Geometry
Stephen Pizer, Kenan Professor
Medical Image Display & Analysis Group
University of North Carolina
This tutorial and other relevant papers can be found at
website: midag.cs.unc.edu
Faculty: me, Ian Dryden, P. Thomas Fletcher,
Xavier Pennec, Sarang Joshi, Carole Twining
MIDAG@UNC
Geometry of Objects in Populations
via representations z
 Uses
for probability density p(z)
 Sampling
p(z) to communicate
anatomic variability in atlases
 Issue:
geometric propriety of samples?
 Log
prior in posterior optimizing
deformable model segmentation =
registration
 Optimizez
p(z|I),
so log p(z) + log p(I|z)
 Or
E(z|I)
MIDAG@UNC
Geometry of Objects in Populations
via representations z
 Uses
for probability density p(z)
 Compare
 Medical
two populations
science
Hypothesis testing with null hypothesis
p(z|healthy) = p(z|diseased)
 If null hypothesis is not accepted, find
localities where probability densities differ
and characterization of shape difference

 Diagnostic:
Is particular patient’s
geometry diseased? p(z|healthy, I)
vs. p(z|diseased, I)
MIDAG@UNC
Needs of Geometric Representation z
& Probability Representation p(z)

Accurate p(z) estimation with limited samples,
i.e., beat High Dimension Low Sample Size
(HDLSS: many features, few training cases)
 Measure
of predictive strength of representation and
statistics [Muller]:
   d zˆ , z
test
k
2
k

 /  d z
training 2
k
test
k
,z

training 2
where “^” indicates projection onto training data principal
space
 Primitives’ positional
correspondence; cases alignment
 Easy fit of z to each training segmentation or image
MIDAG@UNC
Needs of Geometric Representation z
& Probability Representation p(z)
Make significant geometric effects intuitive
 Null probabilities for
geometrically illegal objects
 Localization
 Handle multiple objects
and interstitial regions
 Speed and space

MIDAG@UNC
Schedule of Tutorial







Object representations (Pizer)
PCA, ICA, hypothesis testing, landmark statistics, objectrelative intensity statistics (Dryden)
Statistics on Riemannian manfolds, of m-reps & diffusion
tensors, maintaining geometric propriety (Fletcher)
Statistics on Riemannian manfolds: extensions and
applications (Pennec)
Statistics on diffeomorphisms, groupwise registration,
hypothesis testing on Riemannian manifolds (Joshi)
Information theoretic measures on anatomy,
correspondence, ASM, AAM (Twining)
Multi-object statistics & segmentation (Pizer)
MIDAG@UNC
Representations z of Deformation
 Landmarks
 Boundary
of objects (b-reps)
 Points
spaced along boundary
 or Coefficients of expansion in
basis functions
 or Function in 3D with level set as
object boundary
 Deformation
velocity seq. per voxel
 Medial representation of objects’
interiors (m-reps)
MIDAG@UNC
Landmarks as Representation z
z = (p1, p2, …,pN)
First historically
Kendall, Bookstein,
Dryden & Mardia, Joshi
 Landmarks defined by
special properties
 Won’t find many accurately in 3D
 Global
 Alignment via minimization of
inter-case Spoints distances2


MIDAG@UNC
B-reps as Representation z

Point samples: z = (p1, p2, …,pN)
 Like
landmarks; popular
 Characterization of local translations of shell
 Fit to training objects pretty easy
 Handles multi-object complexes
 Global
 Positional correspondence of primitives

Slow reparametrization optimizing p(z) tightness
 Problems
with geometrically improper fits
 Mesh by adding sample neighbors list

Point, normal samples: z = ([p1,n1],…,[pN,nN])
 Easier
to avoid geometrically improper fits
MIDAG@UNC
B-reps as Representation z
Basis function coefficients
z = (a1, a2, …,aM) with
p(u) = Sk=1M ak k(u)
 Achieves geometric propriety
 Fitting to data well worked out
and programmed
 Implicit, questionable positional
correspondence
 Global,
 Unintuitive
 Alignment via first ellipsoid

1
7
12
Representations via
spherical harmonics
MIDAG@UNC
B-rep via F(x)’s level set: z = F,
an image









Allows topological variability
Topology change
Global
Unintuitive, costly in space
Fit to training cases easy:
F = signed distance to boundary
Modification by geometry limited diffusion
Requires nonlinear statistics: not yet well developed
Serious problems of geometric propriety if stats on F;
needs stats on PDE for nonlinear diffusion
Correspondence?
Localization: via spatially varying PDE parameters??
MIDAG@UNC
Deformation velocity sequence
for each voxel as representation z
 z = ([v1(i.j), v2(i.j),…,vT(i.j)], (i.j)  pixels)
Miller, Christensen, Joshi
 Labels in reference move with deformation
 Series of local interactions
 Deformation energy minimization

 Fluid
flow; pretty slow
Costly in space
 Slow and unsure to fit to
training cases if change
from atlas is large

MIDAG@UNC
M-reps as Representation z
Represent the Egg, not the Eggshell
The eggshell: object boundary primitives
 The egg: m-reps: object interior
primitives
 Poor for object that is tube, slab mix
 Handles multifigure objects and multiobject complexes

 Interstitial
space??
MIDAG@UNC
A deformable model of the
object interior: the m-rep

Object interior primitives:
medial atoms


Local displacement,
bending/twisting, swelling:
intuitive
Neighbor geometry



Objects, figures, atoms, voxels
Object-relative coordinates
Geometric
impropriety:
math check
MIDAG@UNC
Medial atom as a nonlinear
geometric transformation

Medial atoms carry position, width,
2 orientations
deformation T  3 × + × S2
× S2 (× + for edge atoms)
 From reference atom
 Hub translation × Spoke magnification
in common × Spoke1 rotation ×
Spoke2 rotation (× crest sharpness)
 Local

M-rep is n-tuple of medial atoms
 Tn
medial
atom
edge
medial
atom
, n local T’s, a curved, symmetric space
Geodesic distance between atoms
 Nonlinear statistics are required

MIDAG@UNC
Fitting m-reps into training binaries

Optimization penalties
 Distance
between m-rep and
binary image boundaries
 Irregularity penalty: deviation of
each atom from geodesic average
of its neighbors
Yields correspondence(?)
 Avoids geometric impropriety(?)

 Interpenetration

avoidance
Alignment via minimization of
inter-case
Satoms geodesic distances2
MIDAG@UNC
Schedule of Tutorial







Object representations (Pizer)
PCA, ICA, hypothesis testing, landmark statistics, objectrelative intensity statistics (Dryden)
Statistics on Riemannian manfolds, of m-reps & diffusion
tensors, maintaining geometric propriety (Fletcher)
Statistics on Riemannian manfolds: extensions and
applications (Pennec)
Statistics on diffeomorphisms, groupwise registration,
hypothesis testing on Riemannian manifolds (Joshi)
Information theoretic measures on anatomy,
correspondence, ASM, AAM (Twining)
Multi-object statistics & segmentation (Pizer)
MIDAG@UNC
Multi-Object Statistics

Need both



Object statistics
Inter-object relation statistics
We choose m-reps because of
effectiveness in expressing interobject geometry




Medial atoms as transformations of
each other
Relative positions of boundary
Spokes as normals
Object-relative coordinates
MIDAG@UNC
Statistics at Any Scale Level
Global: z
 By object z1k

 Object

By figure (atom mesh) z2k
 Figure

neighbors N(z3k)
By voxel or boundary vertex
 Voxel

neighbors N(z2k)
By atom (interior section) z3k
 Atom

neighbors N(z1k)
neighbors N(z4k)
Designed for HDLSS
atom level
voxel level
relations
MIDAG@UNC
Multiscale models of spatial parcelations
Finer parcellation zj as j increases (scale decreases)
Fuzzy edged apertures zjk, with fuzz (tolerance)
decreasing as j increases
 Geometric representation zjk





We use m-reps to represent objects at moderate scale and
diffeomorphisms to modify that representation at small scale
Level sets of pseudo-distance functions can represent the variable
topology interstitial regions
Provides localization
MIDAG@UNC
Statistics of each entity
in relation to its neighbors at its scale level
on estimating p(zjk , {zjn: n  k}),
via probabilities that reflect both interobject (region) geometric relationship
and object themselves (also for figures)
 Focus
 Markov
random field
 Conditional probabilities p(zjk | {zjn: n  k})
p(zjk | {zjn:  N(zjk)})
 Iterative Conditional Modes – convergence
joint mode of p(zjk , {zjn: n  k} | Image)
=
to
MIDAG@UNC
Representation of multiple objects via
residues from neighbor prediction

Inter-entity and inter-scale relation by
removal of conditional mean of entity
on prediction of its neighbors, then
probability density on residue
| {zjn:  N(zjk)}) = p(zjk  interpoland
zjk: from N(zjk)})
 p(zjk

Restriction of zjk to its shape space
 Early
coarse-to-fine posterior optimization
segmentation results successful, but still under study

Alternative to be explored
 Canonical
correlation
MIDAG@UNC